11 research outputs found
On the local structure of doubly laced crystals
Let be a Lie algebra all of whose regular subalgebras of rank
2 are type , , or , and let be a crystal
graph corresponding to a representation of . We explicitly
describe the local structure of , confirming a conjecture of Stembridge.Comment: 16 pages, 7 figures First version used type crystals; current
version uses type crystals. To appear in J. Comb. Theory, Ser.
Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time
Cellularity of KLR and weighted KLRW algebras via crystals
We prove that the weighted KLRW algebras of finite type, and their cyclotomic
quotients, are cellular algebras. The cellular bases are explicitly described
using crystal graphs. As a special case, this proves that the KLR algebras of
finite type are cellular. As one application, we compute the graded
decomposition numbers of the cyclotomic algebras.Comment: 48 pages, many figures, comments welcom
Applications of Crystal Bases to Current Problems in Representation Theory
This dissertation addresses several current problems in Representation Theory
using crystal bases. It incorporates the results of arXiv:math.QA/0408113 and
arXiv:math.RT/0603547, as well as previously unpublished results